Evan's Reflection on Lockhart

The points Lockhart makes about math and culture in this article were amazing. The analogies to the musician and artist’s dreams were beautiful. I agree that the art of math has been lost due to a societal fascination with utility. This obsession can also be seen in the lack of funding awarded to arts programs. Mathematics is an art who functionality is more widespread, but that does not mean that math and its function are inseparable. I loved Lockhart’s point that marching bands may make armies more efficient, but music exists outside this concept. I also appreciated Lockhart’s elegant definition of math as the art of manipulating simple, imaginary things and seeing how they react.



I was enamoured by Lockhart until he began his critique of “the curriculum,” after which I began to disagree with him more and more. The idea that math does not build upon itself is non-sensical. Even if one were to buy into Lockheart’s idea that students should learn their own math, some concepts will still require students to know the solution to other simpler problems. One cannot imagine the abstract nature of multiplication as groups before understanding the abstract nature of incremental increases. There is no correct way to structure the math curriculum, but I would argue that some structure is necessary. I would use a metaphor of driving from BC to Halifax; sure there are many different roads and paths that you can take, but to contrive of a path that jumps from BC to Ontario to Alberta to NWT to Nova Scotia is nonsensical.



This article relates heavily to Skemp’s idea of instrumental vs relational mathematics. However, Lockhart takes a very radical approach that I do not necessarily agree with. Lockheart seems to argue that not only is instrumental mathematics bad, but teaching it is a moral failing that would make Euclid and Pythagoras roll in their graves. I prefer Skemp’s position of neutrality on this issue, recognizing that there are benefits to both types of instruction.



Moving forward in my teaching career, I will strive to coax the beauty of math out of the hollow shell Lockhart describes as "The Curriculum." I understand that I will not be able to change the curriculum myself, but I believe that by knowing the beauty is there, my excitement and appreciation will be contagious enough to invigorate my students. Hopefully they will learn that math is not the scary beast of boredom, but the beauty of our universe.

My Best and Worst Math Teachers

My best math teacher comes easily to mind. He was my math teacher for grade 11 and grade 12 honors math. In grade 12, I originally had a different teacher. However, I was so unhappy with this new teacher that I begged the administration to switch me into my old teacher’s class. He was one of the most formative male role models in my life. This role modeling existed outside of the classroom. I not only admired him for his intellect in and passion for mathematics, but also for his attitude towards life. I will admit that I was not a morally good person in high school. However, I went through some personal trauma in grade 11 and this force me down a path of self-reevaluation. He recognized this in me and took active measures to guide me down a better path. I would not be as caring or empathetic today if not for him. He educated me in ways well outside the math curriculum.

My worst math teacher was an unorganized mess. Not only were his courses unorganized, but his expectations of us were also unorganized. I felt that I was constantly struggling just to understand where I was, so to understand where I was supposed to go was an impossible task. I missed assignments due to his lack of organization, and even when I handed them in, I missed marks because of his unclear rubrics. It frustrated me and I began doubting my self-efficacy. Like a wet blanket on a fire, he almost extinguished my passion for math.

Response to the locker problem

Instead of looking at all the lockers and using brute force to see a pattern, I looked at a specific locker. I asked “What would cause this locker to be open/closed.” I realized that a locker would be closed if an odd number of people interacted with it, and open if an even amount of people interacted with it.

If each locker was numbered from 1-1000, the number of people interacting with the locker would be equal to the number of factors the locker number has!

If the number of factors is even, the locker is open. If the number of factors is odd, the locker is closed.

This begs the question, which numbers have an odd number of factors? My first thought was prime numbers! Because they would only be visited by their own number. I recognized my mistake quickly; primes are divisible by ONE AND themselves.

I recognized that squares have an odd number of factors, or at least an odd number of distinct factors. For example, 9= 3x3, but 3 is distinct! In more depth, the factors of 9 are 1, 3, 9; an odd number!

I did a quick check by drawing out a list of lockers, and my prediction was confirmed. The square numbers were left closed.

Evan's Reflection on Skemp: Relational vs Instrumental Understanding

 This article has summarized in words ideas that I have had for years. Through my time tutoring students who were failing math, I experienced the two types of understanding. These families wanted a passing grade, and the students wanted instrumental understanding. I would often teach two lessons in parallel, teaching instrumentally first, relationally second. I prefaced the in-depth (what I now know as relational) lesson with “I have taught you the raw ideas of how to do this, now this is how/why the math works.” I provided relational knowledge as a bonus, for them to learn from or tune out if they choose.

 

My favourite part of the article was the analogy to the music lessons. This encapsulated what I have been trying to tell my non-math friends for years. It was such an elegant analogy that my jaw dropped. I cannot wait to share these ideas with my adult friends. It will be the key to make them finally understand why I love a subject that they learned to hate.